FW Math 321, 10/01/2003 Curvilinear Coordinates

FW Math 321, 10/01/2003 Curvilinear Coordinates

Let x, y and z be our usual Cartesian coordinates and consider the general change of coordinates

x = x(q1, q2, q3) (1)
y = y(q1, q2, q3)
z = z(q1, q2, q3)

or, more succinctly, let x = x1, y = x2, z = x3 so

xi = xi(q1, q1, q3), i = 1, 2, 3. (2)

The position vector r can be expressed in terms of the qj’s through the Cartesian expression:

3 (3)

r = xˆ x(q1, q2, q3) + yˆ y(q1, q2, q3) + zˆ z(q1, q2, q3) = xˆi xi(q1, q2, q3).

i=1

Fixing q2, q3 and varying q1, the vector function r = r(q1, q2, q3) describes a curve. The derivative
∂r/∂q1 is tangent to that curve. Likewise, r = r(q1, q2, q3) with q1, q3 fixed and varying q2 gives
another family of curves, one curve for each value of q1, q3, with tangents ∂r/∂q2. Finally, r =
r(q1, q2, q3) with q1, q2 fixed and varying q3 is the third family of curves with tangents ∂r/∂q3. On
the other hand, fixing q1 and varying q2, q3, the vector function r = r(q1, q2, q3) now describes a
family of surfaces, one for each q1, while q2 fixed with varying q1, q3 is another surface and q3 fixed,
varying q1, q2 is the third family of surfaces.

Line Element (Displacement vector) The general line element dr in curvilinear coordinates is
given by (chain rule):

∂r ∂r ∂r 3 ∂r
dr = ∂q1 dq1 + ∂q2 dq2 + ∂q3 dq3 = i=1 ∂qi dqi. (4)

Surface Elements The surface element on the surface q3 = constant, for example, is given by

dA3 = ∂r × ∂r dq1dq2, (5)
∂q1 ∂q2

and likewise for other area elements. An orientation is built into the order of the coordinates.

Volume Element The volume element is given by the mixed (i.e. triple scalar) product

dV = ∂r × ∂r · ∂r dq1dq2dq3. (6)
∂q1 ∂q2 ∂q3

This volume element is positive if q1, q2, q3 correspond to a right-handed frame, negative otherwise.
This important formula, which corresponds to change of variables in multiple integrals, can be
written in the form of a determinant – the Jacobian determinant or simply Jacobian, often denoted
J– which generalizes directly to any dimensions

dV = ∂x/∂q1 ∂x/∂q2 ∂x/∂q3 dq1dq2dq3 = det ∂xi dq1dq2dq3. (7)
∂y/∂q1 ∂y/∂q2 ∂y/∂q3 ∂qj
∂z/∂q1 ∂z/∂q2 ∂z/∂q3

Orthogonal Coordinates

c F. Waleffe, Math 321, 10/01/2003 2

The vectors ∂r/∂qi are key to the coordinates. They provide a natural vector basis. The coordinates
are said to be orthogonal if these tangent vectors are orthogonal to each other. It is useful to define
the unit vector in the qi coordinate direction by

∂r (8)
∂qi = hiqˆi

where hi is therefore

∂r 3 ∂xk 2
hi = || ∂qi || =
k=1 ∂qi . (9)

These hi’s are called the scale or metric factors. The distance traveled in x-space when changing

qi by dqi, keeping the other q’s fixed, is dsi = hidqi.

In curvilinear coordinates, the unit vectors qˆi depend on the coordinates. We need to know their

derivatives with respect to the qj, ∂qˆi/∂qj, for various operations but in particular to determine

the appropriate curvilinear expressions for gradient, divergence and curl. These derivative are

perpendicular to qˆi as, by definition, qˆi · qˆi = 1. Differentiating that expression with respect to qj

yields

qˆi · ∂qˆi = 0. (10)
∂qj

The rate of change of a unit vector is always perpendicular to the unit vector.

Orthogonal coordinates are such that qˆi · qˆj = δij, ∀i, j. We’ll now derive compact expressions for
∂qˆi/∂qj in terms of the qˆi’s and the hi’s, in the case of orthogonal coordinates.
The key relationship arises from the equality of mixed partials

∂2r ∂2r (11)
=

∂qi∂qj ∂qj∂qi

if these 2nd derivatives exist and are continuous, which we assume. Expanding these derivatives in
terms of qˆi and qˆj defined as in (8) gives

qˆi ∂hi + hi ∂qˆi = qˆj ∂hj + hj ∂qˆj . (12)
∂qj ∂qj ∂qi ∂qi

Let 3

∂qˆi = Cijk qˆk, Cijk = qˆk · ∂qˆi . (13)
∂qj ∂qj (14)
k=1

Then (10) implies that

Ciji = 0, ∀i, j

and projecting (12) onto qˆj, with i, j distinct, gives

Cijj = 1 ∂hj , ∀i = j. (15)
hi ∂qi

as qˆj is orthogonal to ∂qˆj/∂qi. Projecting (12) onto qˆk now, with i = k, j = k, gives

hiCijk = hjCjik, ∀i = k, j = k. (16)

c F. Waleffe, Math 321, 10/01/2003 3

Finally, orthogonality of the unit vectors qˆk · qˆi = δik, implies ∂(qˆk · qˆi)/∂qj = 0. Expanding this
out and using the definition of Cijk yields

Cijk = −Ckji, ∀i = k. (17)

The four relationships (14), (15), (16) and (17) determine all the Cijk’s. There are 27 Cijk coeffi-
cients. Eqn. (14) specifies 9 of them and (15) specifies another 6. The remaining 12 coefficients are
determined by the 3 equations given by (16) and the 9 equations (17). Indeed (15) and (17) give

Cjji = −Cijj = −1 ∂hj , ∀i = j, (18)
hi ∂qi

so all coefficients with at least one repeated index, i.e. Ciji, Cijj and Ciik are known. When i, j, k
are all distincts then (16) and (17) give

hiCijk = hj Cjik = −hj Ckij , (19)
hj Cjki = hhCkji = −hkCijk,
hkCkij = hiCikj = −hiCjki.

The first and last column of these equations give the 3-by-3 system

hiCijk + hjCkij = 0,

hkCijk + hj Cjki + = 0, (20)

hiCjki + hkCkij = 0,

whose unique solution is Cijk = Cjki = Ckij = 0. Hence all coefficients with no repeated indices
vanish. The final results are

∂qˆi = 1 ∂hj , ∀i = j (21)
∂qj qˆj hi ∂qi ∀i, j, k (22)

∂qˆi = −qˆj 1 ∂hi − 1 ∂hi , distinct.
∂qi hj ∂qj qˆk hk ∂qk

For orthogonal coordinates, the surface and volume elements can be expressed in terms of the hi’s
as well. For instance, the surface element for the q3 = const. surface is

dA3 = qˆ3 h1h2 dq1dq2, (23)

and the volume element

dV = h1h2h3 dq1dq2dq3, (24)

assuming that q1, q2, q3 is right-handed.
ADD: grad, div, curl... so the h’s determine everything when the q coords are orthogonal. Moreover
they often have simple expressions in terms of the q’s. Let’s look at some important examples.

Cylindrical coordinates x = ρ cos ϕ, y = ρ sin ϕ, z = z. (25)
Spherical coordinates

x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ. (26)

c F. Waleffe, Math 321, 10/01/2003 4
(27)
Elliptical coordinates
x = α cosh u cos v, y = α sinh u sin v, z = z

Non-orthogonal Curvilinear Coordinates

When the coordinates are not orthogonal, the length of the natural basis vectors hi = ||∂r/∂qi|| do
not fully determine the geometry. We need to know all the lengths and all the angles between the
basis vectors, i.e. all the dot-products

gij ≡ ∂r · ∂r = 3 ∂xl ∂xl . (28)
∂qi ∂qj l=1 ∂qi ∂qj

The gij’s are the metric coefficients. Note that gii = h2i .
But there is another important set of basis vectors when the coordinates are non-orthogonal: the

∇qi vectors, i = 1, 2, 3. The ∇qi vector is perpendicular to the qi = constant surface. Therefore
∇q1, for instance, is orthogonal both to ∂r/∂q2 and ∂r/∂q3 which are both tangent to the q1 =
constant surface, but ∇qi is not parallel to ∂r/∂qi unless the coordinates are orthogonal. However,
their dot product is always 1. Indeed, by the chain rule, we have

∂r · ∇qj = 3 ∂xl ∂qj = ∂qj = δij . (29)
∂qi l=1 ∂qi ∂xl ∂qi

When the coordinates are orthogonal, then ∂r/∂qi and ∇qi are parallel and, moreover, their mag-
nitudes are inverse of one another from (29), i.e. ∂r/∂qi = hiqˆi and ∇qi = hi−1qˆi. Again, in the
orthogonal case the hi’s and the qˆi’s determine everything. But in the non-orthogonal case, the
algebra is simpler if we stick with the non-normalized basis vectors ∂r/∂qi and ∇qi, i = 1, 2, 3.
(Actually, it is useful to normalize those basis vectors and define two sets of non-dimensional unit

vectors, as the components of vectors in those normalized bases keep their physical units. One

speaks of “physical curvilinear coordinates” when the normalized bases are used).

Note that ∇qi refers to the gradient of the inverse functions qi = qi(x1, . . . , xn). The implicit func-
tion theorem states that these inverse functions exists provided the Jacobian J ≡ det(∂xi/∂qj)
is not singular. Geometrically, this means that if the vectors ∂r/∂qi, i = 1, . . . , n are not coplanar,
then the qi’s are good coordinates (at least locally) and each point in q-space corresponds to one
and only one point in x-space. The equation (29) specifies that the Jacobian matrices, ∂xi/∂qj
and ∂qi/∂xj are inverses of one another. This is true for their determinant Jx = det(∂xi/∂qj) and
Jq = det(∂qi/∂xj) also: JxJq = 1. The Jacobian determinant (or simply Jacobian) is the ampli-
fication factor of a volume element, i.e. the volume element in q-space: dVq ≡ dq1dq2dq3 becomes
the volume dVx ≡ JxdVq in x-space. Vice-versa: the volume element dVx ≡ dxdydz in x-space
becomes the volume dVq = JqdVx in q-space. We’re back in q-space so the volume should be the
same as originally and that means JxJq = 1. This relation is useful in practice as it may be easier
to compute one Jacobian than the other (example: try the Carnot cycle problem posted in the

exercises). Note that the chain rule also gives

∂xi = δij = 3 ∂xi ∂ql . (30)
∂xj l=1 ∂ql ∂xj

c F. Waleffe, Math 321, 10/01/2003 5

The ∂r/∂qi basis (displacement basis) is particularly appropriate for displacement type vectors
such as the velocity. Indeed the velocity components which are x˙i in Cartesian coordinates are
simply q˙i in the qi coordinates when using the displacement basis as by the chain rule

dr ∂r (31)
= xˆix˙i = q˙j .
dt ∂qj
i j

On the other hand, the ∇qi basis is particularly appropriate for the gradient operator as by the

chain rule again

∇f = ∂f ∂f ∇qj , (32)
xˆi ∂xi = ∂qj
i j

so the components of the gradient are simply ∂f /∂qi in the ∇qi basis, how convenient!
In general, any vector v can be expanded in terms of either basis:

v≡ xˆivi ≡ vj ∂r ≡ v¯j ∇qj. (33)
∂qj
i j j

Using the relationships (29), one obtains the components of one basis by projecting onto the other,

i.e. ∂qi
∂xk
vi = v · ∇qi = vk = v¯j ∇qi · ∇qj , (34)

k j

and ∂r ∂xl ∂r ∂r
∂qi ∂qi ∂qj ∂qi
v¯i = v · = vl = vj · . (35)

l j

These two types of vector components, vi and v¯i, are usually referred to as the “contravariant”
and “covariant” vectors, respectively. They are usually written vj and vj, but I won’t use that
compact but weird notation here. These two sets of components can be directly related to each
other through the metric coefficients

∂r ∂r ∂xl ∂xl , (36)
gij ≡ ∂qi · ∂qj = ∂qi ∂qj
l

and ∂qi ∂qj .
∂xl ∂xl
gij ≡ ∇qi · ∇qj = (37)

l

Indeed from (34) and (35) we have

vi = gij v¯j, (38)

j

and

v¯i = gij vj. (39)

j

The metric coefficients gij and gij specify all the lengths and angles of the bases ∂r/∂qi and ∇qi,
i = 1, 2, 3, respectively. In orthogonal coordinates, we only needed the 3 h’s, but in general we need
the 6 g’s. (The matrices gij and gij are inverse of one another again).

c F. Waleffe, Math 321, 10/01/2003 6

We’re starting to understand why Einstein introduced the summation convention where sums over
repeated indices are implicit, e.g. Einstein simply writes

vj ∂xl ∂xl
∂qi ∂qj

for the double sum ∂xl ∂xl .
∂qi ∂qj
vj

j l

The repeated indices j and l implicitly imply a sum over all values of those indices in Einstein’s
convention. It is a very useful compact notation and you don’t quite need to be Einstein to use it.
However it does require some care and understanding!